Welcome hobophil
When you install TC from a CD image to a drive (HDD or USB) it's important to ensure the system is booting and is also loading extensions from the desired drive, which unfortunately is not intuitive for the new TC user. This is particularly important when you have multiple installations, like a tinycore cd in the cd drive, a USB install, and various HDD installations. Do not make the mistake of thinking tinycore is booting and loading extensions from the drive you selected, because that's not how it works.
The key to a successful and efficient installation is a boot config file which correctly specifies the drive and path to the selected extension store (tce). Not forgetting the importance of making efficient use of both onboot.lst and ondemand features, Unlike the inept onboot.lst file listed in the first post.
Ideally, an efficient onboot.lst file's contents will not include unused extensions or extensions which are dependencies of other extensions, there's a dep file for each extension listing required deps which are loaded when needed, they don't need loading twice...
Regarding the onboot.lst, start with the minimal list of required extensions then add only those extensions required for your hardware which can be found usually within the last 50 line of dmesg output
I'd bet that at least 60% of the extensions listed in your onboot.lst file are not needed
Xorg-7.7.tcz
aterm.tcz
wbar.tcz
wifi.tcz
wl-modules-4.14.10-tinycore.tcz
iw.tcz
pci-utils.tcz
ndiswrapper.tcz
firmware-atheros.tcz
firmware-broadcom_bcm43xx.tcz
firmware-broadcom_bnx2.tcz
firmware-broadcom_bnx2x.tcz
firmware-getB43.tcz
firmware-ipw2100.tcz
firmware-ipw2200.tcz
firmware-iwimax.tcz
firmware-iwl8000.tcz
firmware-iwl9000.tcz
firmware-iwlwifi.tcz
firmware-marvel.tcz
firmware-myri10ge.tcz
firmware-netxen.tcz
firmware-openfwwf.tcz
firmware-ralinkwifi.tcz
firmware-rtl8192ce_se_de.tcz
firmware-rtlwifi.tcz
firmware-ti-connectivity.tcz
firmware-ueagle-atm.tcz
firmware-vxge.tcz
firmware-zd1211.tcz
firmware.tcz
tc-install-GUI.tcz
ezremaster.tcz
nano.tcz
alsa-config.tcz
xf86-video-ati.tcz
firmware-radeon.tcz
firmware-amd-ucode.tcz
graphics-4.14.10-tinycore.tcz
terminus-fonts.tcz
fluxbox.tcz
vlc2.tcz
the above should be reduced tremendously
Start with this, assuming you need the AMD and wl support then check dmesg until all hardware and wifi is working
Xorg-7.7.tcz
aterm.tcz
wbar.tcz
Xlibs.tcz
Xprogs.tcz
fluxbox.tcz
wifi.tcz
wl-modules-4.14.10-tinycore.tcz
pci-utils.tcz
xf86-video-ati.tcz
firmware-radeon.tcz
firmware-amd-ucode.tcz
graphics-4.14.10-tinycore.tcz
alsa.tcz
terminus-fonts.tcz
nano.tcz
you can add vlc2 via ondemand
if you really need wl-modules for a broadcom device don't forget to blacklist the bcma, ssb and b43 modules. then temporarily load firmware-getB43.tcz and run it to install your wifi firmware (if needed)